Simplify the following expression and state the condition under which the simplification is valid. You can assume that $q \neq 0$. $p = \dfrac{3(5q + 2)}{8} \div \dfrac{30q^2 + 12q}{3q} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{3(5q + 2)}{8} \times \dfrac{3q}{30q^2 + 12q} $ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ 3(5q + 2) \times 3q } { 8 \times (30q^2 + 12q) } $ $ p = \dfrac {3q \times 3(5q + 2)} {8 \times 6q(5q + 2)} $ $ p = \dfrac{9q(5q + 2)}{48q(5q + 2)} $ We can cancel the $5q + 2$ so long as $5q + 2 \neq 0$ Therefore $q \neq -\dfrac{2}{5}$ $p = \dfrac{9q \cancel{(5q + 2})}{48q \cancel{(5q + 2)}} = \dfrac{9q}{48q} = \dfrac{3}{16} $